Question: Given that $2^x+ 2^x+ 2^x+ 2^x= 512$, what is the value of $x$?
Answer: First, we simplify the left side, and we have \[2^x+2^x+2^x+2^x = 4\cdot 2^x = 2^2\cdot 2^x = 2^{x+2}.\]Noting that $512 = 2^9$, our equation now is $2^{x+2} = 2^9$, so $x+2 = 9$.  Therefore, $x=\boxed{7}$.